George Kimeldorf

SOME RESULTS ON TCHEBYCHEFFIAN SPLINE FUNCTIONS.

Abstract This report derives explicit solutions to problems involving Tchebycheffian spline functions. We use a reproducing kernel Hilbert space which depends on the smoothness criterion, but not on the form of the data, to solve explicitly Hermite-Birkhoff interpolation and smoothing problems. Sards best approximation to linear functionals and smoothing with respect to linear inequality constraints are also discussed. Some of the results are used to show that spline interpolation and smoothing is equivalent to prediction and filtering on realizations of certain stochastic processes.

POSITIVE DEPENDENCE ORDERINGS

This paper presents a systematic basis for studying orderings of bivariate distributions according to their degree of positive dependence. The general concept of a positive dependence ordering (PDO) is introduced and its properties discussed. Based on this concept, a new ordering of bivariate distributions according to their degree of total positivity of order two (TP2) is presented, and is shown to be a PDO. Properties of this TP2 ordering are derived and numerous applications are presented.

A framework for positive dependence

This paper presents, for bivariate distributions, a unified framework for studying and relating three basic concepts of positive dependence. These three concepts are positive dependence orderings, positive dependence properties and measures of positive dependence. The latter two concepts are formally defined and their properties discussed. Interrelationships among these three concepts are given, and numerous examples are presented.

One-parameter families of bivariate distributions with fixed marginals

For artibrary absolutely continuous univariate distributions F and G we discuss the problem of generating one-parameter families of bivarlate distributions With fixed marginals F and G. According to the statistical literature, for fixed F and G Caere are known co axist only two such families Which satisfy the conditions that they contain the Fee bounds and the distribution corresponding so independent random variables. This paper formalises the problem by suggesting twoadditional conditions these families should satisfy and shows that it is easy to generate many such one-parameter familie In particular, one new family is presented.

Min and max scorings for two-sample ordinal data

Abstract : Ordinal response variables often occur in practice. For example, in clinical trials a subjects response to a drug regime might be categorized as negative, none, fair, or good. There are several common approaches to analyzing two-sample ordinal response data. These procedures applied to the same data can lead to contradictory conclusions. In an attempt to reconcile contradictory results and provide guidance to the practitioner, Kimledorf, Sampson and Whitaker (1992) propose an alternative approach. They find the scores which when assigned to the levels of the ordinal response variable maximize a two-sample test statistic and the scores that minimize that same statistic. Since many of the two-sample statistics are related by monotonic transformations, these extreme scores are in fact extreme scores for several test statistics. Both minimized and maximized test statistics falling into the rejection region clearly indicate a difference between the two populations or treatments. On the other hand if neither of the two extreme statistics fall in the rejection region then no matter what scores are used there will be no significant difference in the two populations. In this paper we review the KsW procedure and its implementation in SAS software.

Silent Duels with Nondiscrete Firing

This paper solves the following class of zero-sum two-person games, which are closely related to the classical discrete-fire duels. For

Dependence structures in which uncorrelatedness implies independence

i = 1

Some results on the maximal correlation in 2 × k contingency tables

and 2, associated with Player i is a finite, nonnegative number

On Nash equilibrium points and games of imperfect information

M_i

Optimized scorings for ordinal data for the general linear model

and a strictly increasing function